For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

Calculus, more properly called analysis (or real analysis, or in older literature, infinitesimal analysis), is the branch of mathematics studying the rate of change of quantities, which can be interpreted as slopes of curves, and the lengths, areas and volumes of objects. Calculus is sometimes divided into differential and integral calculus, which are concerned with derivatives

$$\frac{dy}{dx}= \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$

and integrals

$$\int_a^b f(x)\,dx = \lim_{\Delta x \to 0} \sum_{k=0}^n f(x_k)\ \Delta x_k,$$

respectively. The Fundamental Theorem of Calculus relates these two concepts.

While ideas related to calculus were known for some time (Archimedes' method of exhaustion was a form of calculus), it was not until the independent work of Newton and Leibniz that the modern elegant tools and ideas of calculus were developed. Riemann and Lebesgue later extended the ideas of integration. More recently, the Henstock–Kurzweil integral has led to a more satisfactory version of the second part of the Fundamental Theorem of Calculus.

Even so, many years elapsed until mathematicians such as Cauchy and Weierstrass put the subject on a mathematically rigorous footing; it was Weierstrass who formalized the definition of continuity of a function, proved the intermediate value theorem, and proved the Bolzano-Weierstrass Theorem.

Source: Wolfram Mathworld